nLab
bifibration

Contents

Definition

A bifibration of categories is a functor

EB E \to B

that is both a Grothendieck fibration as well as an opfibration.

Therefore every morphism f:b 1b 2f : b_1 \to b_2 in a bifibration has both a push-forward f *:E b 1E b 2f_* : E_{b_1} \to E_{b_2} as well as a pullback f *:E b 2E b 1f^* : E_{b_2} \to E_{b_1}.

Examples

Relation to monadic descent

Ordinary Grothendieck fibrations correspond to pseudofunctors to Cat, by the Grothendieck construction and hence to prestacks. For these one may consider descent.

If the fibration is even a bifibration, there is a particularly elegant algebraic way to encode its descent properties; this is monadic descent. The Benabou–Roubaud theorem characterizes descent properties for bifibrations.

Relation to distributive laws

Fibrations and opfibrations on a category CC (or more generally an object of a suitable 2-category) are the algebras for a pair of pseudomonads. If CC has pullbacks, there is a pseudo-distributive law between these pseudomonads, whose joint algebras are the bifibrations satisfying the Beck-Chevalley condition; see (von Glehn).

Remark

A bifibration F:EBF:E\to B such that F op:E opBF^{op}:E^{op}\to B is a bifibration as well is called a trifibration (cf. Pavlović 1990, p.315).

References

Revised on June 16, 2017 12:05:23 by Mike Shulman (76.167.222.204)